Ddual -- holonomic dual of a D-module

Synopsis

Usage:

Ddual M, Ddual I
Inputs:
- M, a module, over the Weyl algebra D
- I, an ideal, which represents the module M = D/I
Outputs:
- a module, the holonomic dual of M

Description

If M is a holonomic left D-module, then Extⁿ_D(M,D) is a holonomic right D-module. The holonomic dual is defined to be the left module associated to Extⁿ_D(M,D). The dual is obtained by computing a free resolution of M, dualizing, and applying the standard transposition to the n-th homology.

i1 : I = AppellF1({1,0,-3,2})

               3  2    2           2  2                 2             
o1 = ideal (- x Dx  - x y*Dx*Dy + x Dx  + x*y*Dx*Dy - 2x Dx + 2x*Dx, -
     ------------------------------------------------------------------------
        2         3  2                2  2              2
     x*y Dx*Dy - y Dy  + x*y*Dx*Dy + y Dy  + 3x*y*Dx + y Dy + 2y*Dy + 3y,
     ------------------------------------------------------------------------
     x*Dx*Dy - y*Dx*Dy + 3Dx)

o1 : Ideal of QQ[x, y, Dx, Dy]

i2 : Ddual I

o2 = cokernel | 0  xDy-yDy-4 x2Dx+y2Dy-xDx-yDy+x+4y y2DxDy+y2Dy^2-yDxDy-yDy^2+4xDx+4yDx+5yDy-4Dx+4 0                           |
              | Dx -yDy-1    0                      0                                              y3Dy^2-y2Dy^2+7y2Dy-2yDy+5y |

                                                            2
o2 : QQ[x, y, Dx, Dy]-module, quotient of (QQ[x, y, Dx, Dy])

Caveat

The input module M should be holonomic. The user should check this manually with the script Ddim.

Ways to use Ddual:

Ddual(Ideal)
Ddual(Module)

For the programmer